How to merge two identical tables that have a primary key?

This seems like a simple task, but I am struggling to find a way to do this. I have two tables (A and B) with the same structure. Both have an auto-increment primary key. I want the data from table B to go in to table A. I tried:

insert into A select * from B

However I get an error that a primary key already exists. I would like the new rows from table B to get a new primary key when I insert them in to table A - so basically discard the primary key from B but insert all the other columns. Seems like it should be a simple query but I can't figure it out. Thank you.